Dr. Caucao-Paillán, Sergio

We consider a Banach spaces-based mixed variational formulation recently proposed for the stationary u(I)-rheology model of granular materials, and develop the first reliable and efficient residual-based a posteriori error estimator for its associated mixed finite element scheme in both 2D and 3D, considering PEERS and AFW-based discretizations. For the reliability analysis, and due to the nonlinear nature of the problem, we employ the first-order Gâteaux derivative of the global operator involved in the problem, combined with appropriate small-data assumptions, a stable Helmholtz decomposition in nonstandard Banach spaces, and local approximation properties of the Raviart–Thomas and Clément interpolants. In turn, inverse inequalities, the localization technique based on bubble functions in local Lp-spaces, and known results from previous works are the main tools yielding the efficiency estimate. Finally, several numerical examples confirming the theoretical properties of the estimator and illustrating the performance of the associated adaptive algorithms are reported. In particular, the case of fluid flow through a 2D cavity with two circular obstacles is considered.

We propose and analyze a new mixed finite element method for the nonlinear problem given by the stationary convective Brinkman–Forchheimer equations. In addition to the original fluid variables, the pseudostress is introduced as an auxiliary unknown, and then the incompressibility condition is used to eliminate the pressure, which is computed afterwards by a postprocessing formula depending on the aforementioned tensor and the velocity. As a consequence, we obtain a mixed variational formulation consisting of a nonlinear perturbation of, in turn, a perturbed saddle point problem in a Banach spaces framework. In this way, and differently from the techniques previously developed for this model, no augmentation procedure needs to be incorporated into the formulation nor into the solvability analysis. The resulting non-augmented scheme is then written equivalently as a fixed-point equation, so that recently established solvability results for perturbed saddle-point problems in Banach spaces, along with the well-known Banach–Nečas–Babuška and Banach theorems, are applied to prove the well-posedness of the continuous and discrete systems. The finite element discretization involves Raviart–Thomas elements of order for the pseudostress tensor and discontinuous piecewise polynomial elements of degree for the velocity. Stability, convergence, and optimal a priori error estimates for the associated Galerkin scheme are obtained. Numerical examples confirm the theoretical rates of convergence and illustrate the performance and flexibility of the method. In particular, the case of flow through a 2D porous media with fracture networks is considered.